Problem 14 - Algebra 2 and Trigonometry

Question

In \(11-18, P\) is a point on the terminal side of an angle in standard position with measure \(\theta\) and on a circle with center at the origin and radius \(r .\) For each point \(P,\) find: a. \(r\) b. \(\csc \theta\) c. \(\sec \theta\) d. \(\cot \theta\) \((-5,-5)\)

Step-by-Step Solution

Answer

a. \(r = 5\sqrt{2}\), b. \(\csc \theta = -2\sqrt{2}\), c. \(\sec \theta = -2\sqrt{2}\), d. \(\cot \theta = 1\).

Step 1: Calculate the Radius (r)

The point \((-5, -5)\) lies on the terminal side of the angle \(\theta\) in a circle centered at the origin. The radius \(r\) is the distance from the origin \((0,0)\) to the point \((-5,-5)\). Use the distance formula: \[ r = \sqrt{(-5-0)^2 + (-5-0)^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2} \] Thus, the radius \(r\) is \(5\sqrt{2}\).

Step 2: Calculate \(\sin \theta\)

The definition of \(\sin \theta\) in standard position is the \(y\)-coordinate of the point divided by the radius \(r\). Thus: \[ \sin \theta = \frac{-5}{5\sqrt{2}} = -\frac{1}{\sqrt{2}} \] Simplify: \[ \sin \theta = -\frac{\sqrt{2}}{2} \]

Step 3: Calculate \(\csc \theta\)

The cosecant \(\csc \theta\) is the reciprocal of \(\sin \theta\). Therefore, \[ \csc \theta = \frac{1}{\sin \theta} = \frac{1}{-\frac{\sqrt{2}}{2}} = -2\sqrt{2} \]

Step 4: Calculate \(\cos \theta\)

The definition of \(\cos \theta\) in standard position is the \(x\)-coordinate of the point divided by the radius \(r\). Thus: \[ \cos \theta = \frac{-5}{5\sqrt{2}} = -\frac{1}{\sqrt{2}} \] Simplify: \[ \cos \theta = -\frac{\sqrt{2}}{2} \]

Step 5: Calculate \(\sec \theta\)

The secant \(\sec \theta\) is the reciprocal of \(\cos \theta\). Therefore, \[ \sec \theta = \frac{1}{\cos \theta} = \frac{1}{-\frac{\sqrt{2}}{2}} = -2\sqrt{2} \]

Step 6: Calculate \(\tan \theta\)

The tangent \(\tan \theta\) is the ratio of \(\sin \theta\) to \(\cos \theta\). Thus: \[ \tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1 \]

Step 7: Calculate \(\cot \theta\)

The cotangent \(\cot \theta\) is the reciprocal of \(\tan \theta\). Therefore, \[ \cot \theta = \frac{1}{\tan \theta} = 1 \]

Step 8: Conclusion

With all calculations completed, the values are: - a. Radius \(r = 5\sqrt{2}\).- b. \(\csc \theta = -2\sqrt{2}\).- c. \(\sec \theta = -2\sqrt{2}\).- d. \(\cot \theta = 1\).

Key Concepts

Radius Calculation

In understanding trigonometric relations, radius calculation is a fundamental step. When a point lies on the terminal side of an angle in standard position on a circle centered at the origin, it forms a right triangle where the radius is essential. You can picture this as the hypotenuse of the triangle formed between the origin, the x-axis, and the point itself. To find this radius, use the distance formula:\[ r = \sqrt{(x-0)^2 + (y-0)^2} \]For example, given point \((-5, -5)\), the radius calculated becomes:- Calculate x and y differences: - \((-5 - 0)^2\) = 25 - \((-5 - 0)^2\) = 25- Sum and simplify: - \(r = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}\) The computed radius helps derive various trigonometric ratios relevant to our further calculations.

Cosecant Function

Once the radius is known, the trigonometric function cosecant, denoted as \(\csc \theta\), is straightforward to compute. The cosecant function is simply the reciprocal of the sine function:- Definition: - \(\csc \theta = \frac{1}{\sin \theta}\)Given that sine is identified by the ratio of the y-coordinate to the radius, using the point \((-5, -5)\), we observe:- Calculate \(\sin \theta\): - \(\sin \theta = \frac{-5}{5\sqrt{2}} = -\frac{1}{\sqrt{2}} = -\frac{\sqrt{2}}{2}\)From this, obtain \(\csc \theta\):- Take reciprocal: - \(\csc \theta = \frac{1}{-\frac{\sqrt{2}}{2}} = -2\sqrt{2}\)The cosecant is particularly useful in various mathematical applications, providing precise relationships for angles in geometry and trigonometry.

Secant Function

The secant function, symbolized by \(\sec \theta\), is another vital trigonometric ratio derived from the cosine function. It embodies the reciprocal of cosine:- Definition: - \(\sec \theta = \frac{1}{\cos \theta}\)Cosine evaluates to the x-coordinate over the radius. For point \((-5, -5)\):- Compute \(\cos \theta\): - \(\cos \theta = \frac{-5}{5\sqrt{2}} = -\frac{1}{\sqrt{2}} = -\frac{\sqrt{2}}{2}\)Evaluate \(\sec \theta\) by inversion:- Reciprocal calculation: - \(\sec \theta = \frac{1}{-\frac{\sqrt{2}}{2}} = -2\sqrt{2}\)Secant provides insights into the extension of distances in triangles and serves heavily in real-world scenarios, such as projectile motion and oscillations.

Cotangent Function

Cotangent, denoted as \(\cot \theta\), is equally significant and is the reciprocal of the tangent function. It offers a different viewpoint by relating cosine to sine:- Definition: - \(\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}\)Given the earlier values for \(\cos \theta\) and \(\sin \theta\) for the point \((-5, -5)\):- Find the tangent first: - \(\tan \theta = \frac{\sin \theta}{\cos \theta} = \frac{-\frac{\sqrt{2}}{2}}{-\frac{\sqrt{2}}{2}} = 1\)From this, deduce \(\cot \theta\):- Calculate the reciprocal: - \(\cot \theta = \frac{1}{1} = 1\)Cotangent serves imperative functions in understanding phase relationships and periodic behavior in mathematics and various scientific fields.